# Proof that All Triangles can be Inscribed into a Circle

Fellow Mathers, I am currently in a Euclidean Geometry class. We are using the SMSG axioms to build our geometry. I am struggling in proving the fact that all triangles can be inscribed into a circle, i.e. that...

$$Ɐ$$ triangles $$ΔABC$$, $$∃$$ a circle $$Θ$$ such that $$A,B,C ∈ Θ$$.

Anyone who can provide a proof or point me in the right direction, it would be greatly appreciated. - SDH

• This question may help: math.stackexchange.com/questions/16634/… – David Peterson Sep 15 at 4:55
• In the title "cyclic" is misleading. Periphrasing into "can be inscribed in a circle" would be probably better. – Jean Marie Sep 15 at 5:49
• @JeanMarie Will do. Don’t really know what I’m talking about with the jargon, just saw that terms being used online. – SDH Sep 15 at 5:51

Let be $$D$$ the intersection point of the perpendicular bisectors of $$AC$$ and $$BC$$. You have that
1. $$AD=CD$$ since $$D$$ belongs to $$s$$
2. $$CD=BD$$ since $$D$$ belongs to $$r$$ Therefore $$AD=BD=CD=R$$ and the circle of center $$D$$ and radius $$R$$ is the required circle.